In the case of \(\mathcal {T}=\mathbb {R}^d\)  it is shown in [1]} that a measure-valued process with generator \(Q_n\)  exists, and is in fact a jump process in \(\Gamma \)  corresponding to the jump kernel \(\bar{\kappa }_n\)  shown below. However, for our general setting with \(\mathcal {T}\)  a compact Polish space, we will take (REF ) simply as a starting point, and do not consider the existence or convergence of the measure-valued process \(\nu _t^n\)  itself—even though we will sometimes borrow the language of jump processes for illustration purposes.
